Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number of disjoint $S_r^d$ that can be completely contained in $S_R^d$?

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Although the book "Sphere packings, lattices and groups" by Conway and Sloane deals mostly with infinite packings, it has an extensive bibliography that gives some references for the problem you're interested in too. Here are some entries that looked relevant, although I haven't read them:

Gritzmann and Wills, "Finite packing and covering", in Gruber and Wills, "Handbook of Convex Geometry"

Wills, "Finite sphere packings and sphere coverings"

Wills, "Finite sphere packings and the methods of Blichfeldt and Rankin"

Fejes Toth, "Packing and covering", in Goodman and O'Rourke, "Handbook of Discrete and Computational Geometry"

Nuermela and Ostergard, "Dense packings of congruent circles in a circle"